Take N = 3. As remarked above, we need only consider a two-period model. We will take the binomial model, and let Q be the probability measure that assigns equal probability of 1/4 to each of the four elements in Ω = {UU, UD, DU, DD}. Define the random variables v(0, 1), v(1, 2) and v(2, 3) by v(0, 1) = 0.7, v(1, 2)(U) = 0.8 v(1, 2)(D) = 0.6. v(2, 3)(UU) = 0.9, v(2, 3)(UD) = 0.7, v(2, 3)(DU) = 0.7, v(2, 3)(DD) = 0.5.

(a) Use (20.28) to find the distribution under Q of the other random variables v(k, n) for k n, which will make the market arbitrage-free.

(b) Show that it is not necessarily true that v(k, n) = v(k, m)Ekv(m, n) if m ≠ k + 1.